Positive and compacton-type solutions for a quasilinear two-point boundary value problem

Abstract

In this paper we investigate a one-dimensional p-Laplacian problem, whose nonlinearity comprises both p-sublinear and p-superlinear terms. By studying the multiplicity of solutions to an auxiliary problem, we show how convenient choices of the parameters involved yield the existence of at least three classical positive solutions satisfying Hopf's boundary condition. In addition, we provide the existence of two distinct curves of parameters along which our problem admits at least a compacton-type solution.

Uniqueness of positive and compacton-type solutions for a resonant quasilinear problem

“…For all i ∈ N ∪ {0}, F (i) stands for the i-th derivative of F where, as customarily, F (0) def = F . Invoking Lemmas 2.1 and 2.2 of [1], one can easily prove the following properties.…”

“…Our approach relies on quadrature methods and was successfully adopted in the recent paper [1] to deal with a problem similar to (P µ ) with p-superlinear and p-sublinear terms. In that case the competition between the opposite trends of the nonlinearity resulted in the existence of (at least) three positive solutions and two distinct curves of compactons (Theorems 2.7 and 2.9 of [1], respectively). Here the situation is more delicate as the interaction occurs between a p-linear resonant term and two p-sublinear ones.…”

“…The properties of the following function will be often used in the next computations (cf. [1,Lemma 2.3]).…”